Polinomio Homogeneus – Wikipedia, free encyclopedia

In mathematics, a homogeneous polynomial It is a polynomial in which each of its terms (monomials) have the same degree; or its elements are of the same dimension. For example,

${displaystyle x^{5}+2x^{3}y^{2}+9x^{1}y^{4}}$

It is a homogeneous polynomial of grade 5, in two variables; The sum of the exponents is always 5.

A algebraic form , it simply form It is another name for a homogeneous polynomial. A homogeneous grade 2 polynomial is a quadratic form, and can be represented as a symmetrical matrix. The theory of algebraic forms is very extensive, and has numerous applications in all other mathematics and theoretical sciences.

Symmetrical tensioners [ To edit ]

Homogeneous polynomials in a vector space can be built directly from symmetrical tensioners, and vice versa. For defined vector spaces on the bodies of real or complex numbers, the homogeneous polynomial system and symmetrical tensioners are in fact isomorphic. This relationship is usually expressed as follows.

Being X It is AND vector vectors, and T The multilineal map or symmetrical tensioner:

${displaystyle {begin{matrix}T:&underbrace {Xtimes Xtimes cdots times X} &to &Y\&n&&\end{matrix}}}$

Diagonal operator is defined

${displaystyle Delta }$

as:

${displaystyle {begin{matrix}Delta :&X&to &X^{n}\&x&mapsto &(x,x,dots ,x)\end{matrix}}}$

The homogeneous polynomial

${displaystyle {widehat {T}}}$

of degree n associate with T It is simply

${displaystyle {widehat {T}}=Tcirc Delta }$

, so that

${displaystyle {widehat {T}}(x)=(Tcirc Delta )(x)=T(x,x,ldots ,x)}$

Written in this way, it is clear that a homogeneous polynomial is a homogeneous degree function n . This, for a scalar a , one has

${displaystyle {widehat {T}}(ax)=a^{n}{widehat {T}}(x)}$

Inversely, given a homogeneous polynomial

${displaystyle P}$

, one can build the corresponding symmetrical tensioner

${displaystyle {check {P}}}$

, which immediately follows a multilinearity of the tensor through a polarized formula:

${displaystyle {check {P}}(x_{1},x_{2},cdots x_{n})={frac {1}{2^{n}n!}}sum _{varepsilon _{i}=pm 1 atop 1leq ileq n}varepsilon _{1}varepsilon _{2}cdots varepsilon _{n}Pleft(sum _{i=1}varepsilon _{i}x_{i}right)}$

${displaystyle {mathcal {L}}(X^{n},Y)}$

denotes the space of symmetrical tensioners of rank n , and

${displaystyle {mathcal {P}}(X,Y)}$

Denotes the homogeneous grade space space n . If the space vector X It is AND They are above the real or complex numbers (or more generally, on top of a zero characteristic body), then those two spaces are isomorphic, with the mapping given by hats and we verify:

${displaystyle {widehat {;}}:{mathcal {L}}(X^{n},Y)to {mathcal {P}}(X,Y)}$

and

${displaystyle {check {;}}:{mathcal {P}}(X,Y)to {mathcal {L}}(X^{n},Y)}$

Algebraic form [ To edit ]

Algebraic form, or simply form, is another term for homogeneous polynomials. These are generally used for quadratic forms of degrees 3 and more, and in the past they were also known as how many . By specifying the type of form, one has to give your grade in one way, and the number of variables n . One is on of some field K given, if this goes from K ⁿ a K , where n It is the number of variables of the form.

A form above some field K in n variables represents 0 If there is an element

( x _first , …, x _n )

in K ⁿ similar that at least

x _i ( i = 1, …, n )

It is not equal to zero.

Basic properties [ To edit ]

The number of different homogeneous monomials m in n variables is

${Displaystyle {frac {(m+n-1)!} {m! (n-1)!}}}}$

The Taylor series of a homogeneous polynomial P expanded to the point x can be written as

${DISPLAYSTYLE {BEGIN {MATIX} P (x + y) = SM = 0} {p} {p} {, x} & underbrace} y, y, dots, y}).} \ & J & N-J \ EN \ ENDICX}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$

Another useful identity is

${displaystyle {begin{matrix}P(x)-P(y)=sum _{j=0}^{n-1}{n choose j}{check {P}}(&underbrace {y,y,dots ,y} &underbrace {(x-y),(x-y),dots ,(x-y)} ).\&j&n-j\end{matrix}}}$

Homogeneous polynomial had an important role in nineteenth -century mathematics.

The two obvious areas where it could be applied were projective geometry, and numbers theory (to a lesser extent). Geometric use was related to invariant theory.